Following a recent discussion with an EE (who declared the speed of light to be an invariant) I wondered how many electrical engineers understand the important difference.

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I'm a EE, and I can't say I've ever had the need or desire to make the distinction. In general usage there isn't much difference, but maybe there is for math/physics context.

The only important difference I can think of is that the term invariant often applies to an entity that is unchanged by a transformation rule, and maybe constant means unchanging always, implying with respect to time?

With regards to the speed of light, I would tend to say that the speed of light is a constant, at least based on usual physics. Although there are some theories that speculate that the speed of light changes slowly over time, in which case it would not be.

I wouldn't be bothered that someone called the speed of light invariant, since we can talk about reference frame transformations and note that the speed of light remains invariant to these transformations, while the speed of any solid object changes when the reference frame changes.

I'm sure you're going to tell me I'm wrong, but I thought I'd stick my neck on the chopping block, just for the sake of learning.

I've been a EE for 45 years and I think it is a distinction without a difference, sort of like asking what is the difference between a constant and a random variable with a finite mean and a zero variance.

As I understand it the speed of light does change, depending somewhat on its environment. Glass, for example, causes the speed to change, and is different according to frequency (hence prisms). I understand TWT do something similar, though subtler. Ditto for going through wire in antennas.

In mathematics we distinguish between the variables and constants of an equation.

eg in y = ax y and x are variables - a is a constant.

In physics we assign meaning to the variables and sometimes also to the constants.
The equations now describe different states of a system, where the variables do just that and the constants are, well constant.

The equations may be said to describe simple transformations.

We even formally collect bunches of these variables to form complex (complicated) objects such as the momentum vector or eelctric stress tensor or whatever.

This allows us to manipulate more difficult transformations.

A transformation represents two views of the same system either before and after some process or concurrently from different viewpoints.

Often the variables will interact during a complex transformation. We handle this by generalising our simple equation above to a matrix equation

Y = A X

or in a simple example (sorry I can't do tex matrices)

y1 1 0 x1 = x1
y2 1 1 x2= x1+x2

A = 1 0
......0 1 if someone could explain how to tidy up the tex I will replace.

Under this transformation variable x1 is Invariant, but x2 is not.
The constants involved are of course 1 and 0

The speed of light is a constant, not a variable and not manipulable by us.

Equally we physicists search for quantities (variables) like x1 that are invariant under the given transformation as they are of special interest.

One such is electric charge which is invariant under all mechanical transformations, unlike mass, length and time!

It's largely a matter of the semantics in the particular technical area.

For example, in math, most would agree that 3 and 7.2 are constants. If one has an equation f(x) = 0, then one refers to the quantity x as a variable, as its value comes from a finite or infinite set of constants. If one is only interested in the value(s) of x that make the equation true, then one might call those constants the root(s) of the equation.

In physics, definitions change over time (meaning they're not constant ). In studiot's example, the speed of light is currently defined to be a constant. Note, however, that for many decades it was an experimentally determined value that changed slightly over the years as different experimenters got slightly different values. For the last few decades it has been in vogue to consider it a constant, most likely because it being a constant is one of the two fundamental postulates of special relativity, which is a physical theory that has withstood a large variety of experimental tests. The wikipedia page also says that most of the measurement uncertainty of the speed of light was traced to the uncertainty in the length of the meter. Regardless, there is no a priori reason why it has to be a constant nor that it has been constant over time. Only experimental measurements can verify that. Similar considerations apply to the fundamental charge, which hasn't been an experimental constant historically as humorously related by Feynman.

Invariants are things that don't change when one changes one's viewpoint. They're invariantly useful (pun intended). For example, in programming, a good programmer will know the invariants of an algorithm and use them to check that his code hasn't changed the invariant (and thus it serves as a useful check against bugs). In math, often a particular function (or functional) should be invariant regardless of what coordinate system you use to evaluate it. For example, a line integral over a path should yield the same value whether you use Cartesian or spherical coordinates. The modern parlance is that the invariant doesn't depend on what chart(s) you put on the manifold.

Invariants in physics are quite important and are often deeply related to the symmetries of a system. These can be discussed using the machinery of Noether's theorem or the work Lie did on the symmetries of differential equations in the 1800's. In a nutshell, the symmetries correspond to "constants of the motion". These, in turn, get related to the integration constants you learned about when (if) you studied differential equations. An example in mechanics is that if a certain function of coordinates and momenta (the "Hamiltonian") doesn't depend explicitly on time, then the total energy of the system is a constant. There's thus a deep connection between the constants and the points of view (i.e., the transformations).

I was recently cleaning out the garage and came across some work I did when I was a student in the early 70's on this symmetry stuff. The stuff is gibberish to me now, but at the time my teacher and I thought it was pretty neat. We were able to do things like show that the n dimensional free particle and n dimensional isotropic harmonic oscillator had symmetry groups that could be related through a number of different transformations -- and even integrate some of the local infinitesimal transformation generators to get the (rather complicated) finite transformations. They were all representations of the same SL(n+2, R) transformation group, so, in some sense, these physical systems were equivalent. I look at the complicated transformations now and wonder if they've ever been of any practical use to anyone (certainly not to me, as I pretty much ignored that stuff after I got out of school).

Based on your example above, in the math context, a constant is taken from the set of all integers or all real number. The term invariant is applied to a number in the context of some transformation process.

The transform of a constant is not always an invariant, while the transform of an invariant is alway a constant.

As in all discussions, the terms need to be precisely defined (and agreed upon by the participants) before higher level discourse can take place. Otherwise, we all redouble our efforts after losing sight of our goals and I'm sure all of us have been in those situations.

So, studiot, I'd appreciate hearing your definitions of a "constant" and an "invariant". It will be interesting to see to what extent others use the same definitions.

Obviously I didnt do it very well so I will have another go. I will try to take in Bills query as I go.

Both constants and invariants are basically mathematical ideas. When physicists apply mathematics to the real world they may add meaning ( and sometimes restrictions) not present in the original mathematical ideas.

A simple mathematical entity is an expression.

(3 tan 2x ) and (2 tan 3x ) are both expressions and different from each other.

Such expressions are formed by applying coefficients which we cant alter to variables, which we can. Hence we call the coefficients constants.

It is these coefficients, or constants, which define and distinguish expressions, not the variables.

We could substitute another variable (y, z, t etc) for x or allocate any value within the domain and the expression would be the same.

If, however, we changed any coefficient we would have a different expression.

Such are constants then - they are the coefficients which define mathematical expressions.

Mathematical expressions do not, of themselves, possess invariants.

Invariants arise when we consider mathematical processes. Such processes may apply to expressions with coefficients as above or to other mathematical entities such as shapes.

An operand of a process (the object being processed) is called an invariant if it is unchanged at the end of the process.

So, for instance, an equilateral triangle with its base on the x axis and apex on the y axis is unchanged or invariant under the transformation process of reflection in the y axis.
Note that in this example there is not a coefficient or constant to be seen.

When we consider transformations of mathematical expressions it is the variable that is the object being processed and the variable that is considered an invariant if it is unchanged by the transformation.

So for instance exp(x) is invariant under the process of integration.

This is as far as the mathematical definitions can take matters. As I previously observed physicists add meaning to the mathematical objects. I will deal with the additional layer(s) of structure imposed by applications to physics in a subsequent post.

There is definitely semantics at play here. Invariant, like many other words, has a somewhat different meaning based on its context.

However, in the context of the subjects discussed in this forum, invariant means unaffected by a particular transformation. One could certainly find any number of physical models for which there are transformations wherein variables, or indeed entire functions, are invariant under those transformations, but they are not constants.

When used as a noun, invariant has only one meaning as far as I know. However, despite the root word, variant, being only a noun, several dictionaries I checked, both on-line and printed, list invariant as both an adjective and a noun.

Noun1.invariant - a feature (quantity or property or function) that remains unchanged when a particular transformation is applied to it characteristic, feature - a prominent attribute or aspect of something; "the map showed roads and other features"; "generosity is one of his best characteristics"

math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangementAdj.1.invariant - unaffected by a designated operation or transformation math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

Adj.1.invariant - unaffected by a designated operation or transformation math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

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Not sure I follow your dictionary extracts.

The emboldened text certainly refers to nouns not adjectives, unless they have changed their spots in the swim across the Atlantic.